A flying arrow occupies exactly one position at any moment. Thus, at every instant of time, the arrow is at rest. But then the motion of the arrow is impossible, since time is composed of such (motionless) instants. In other words, something that is always at rest cannot be in motion.
The paradox is baffling at the very least, until Veronese's conception of the continuum comes to the rescue. Although instants (points) of time do exist, with exactly one position of the arrow belonging to each of them, time itself is not composed of instants. Time can only be decomposed into intervals of non-zero length, and in none of those is the arrow at rest.
This is a pure mathematical resolution of the paradox. No need to resort to physics.